// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009 Claire Maurice
// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_COMPLEX_SCHUR_H
#define EIGEN_COMPLEX_SCHUR_H

#include "./HessenbergDecomposition.h"

namespace Eigen {

namespace internal {
    template <typename MatrixType, bool IsComplex> struct complex_schur_reduce_to_hessenberg;
}

/** \eigenvalues_module \ingroup Eigenvalues_Module
  *
  *
  * \class ComplexSchur
  *
  * \brief Performs a complex Schur decomposition of a real or complex square matrix
  *
  * \tparam _MatrixType the type of the matrix of which we are
  * computing the Schur decomposition; this is expected to be an
  * instantiation of the Matrix class template.
  *
  * Given a real or complex square matrix A, this class computes the
  * Schur decomposition: \f$ A = U T U^*\f$ where U is a unitary
  * complex matrix, and T is a complex upper triangular matrix.  The
  * diagonal of the matrix T corresponds to the eigenvalues of the
  * matrix A.
  *
  * Call the function compute() to compute the Schur decomposition of
  * a given matrix. Alternatively, you can use the 
  * ComplexSchur(const MatrixType&, bool) constructor which computes
  * the Schur decomposition at construction time. Once the
  * decomposition is computed, you can use the matrixU() and matrixT()
  * functions to retrieve the matrices U and V in the decomposition.
  *
  * \note This code is inspired from Jampack
  *
  * \sa class RealSchur, class EigenSolver, class ComplexEigenSolver
  */
template <typename _MatrixType> class ComplexSchur
{
public:
    typedef _MatrixType MatrixType;
    enum
    {
        RowsAtCompileTime = MatrixType::RowsAtCompileTime,
        ColsAtCompileTime = MatrixType::ColsAtCompileTime,
        Options = MatrixType::Options,
        MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
        MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
    };

    /** \brief Scalar type for matrices of type \p _MatrixType. */
    typedef typename MatrixType::Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real RealScalar;
    typedef Eigen::Index Index;  ///< \deprecated since Eigen 3.3

    /** \brief Complex scalar type for \p _MatrixType. 
      *
      * This is \c std::complex<Scalar> if #Scalar is real (e.g.,
      * \c float or \c double) and just \c Scalar if #Scalar is
      * complex.
      */
    typedef std::complex<RealScalar> ComplexScalar;

    /** \brief Type for the matrices in the Schur decomposition.
      *
      * This is a square matrix with entries of type #ComplexScalar. 
      * The size is the same as the size of \p _MatrixType.
      */
    typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrixType;

    /** \brief Default constructor.
      *
      * \param [in] size  Positive integer, size of the matrix whose Schur decomposition will be computed.
      *
      * The default constructor is useful in cases in which the user
      * intends to perform decompositions via compute().  The \p size
      * parameter is only used as a hint. It is not an error to give a
      * wrong \p size, but it may impair performance.
      *
      * \sa compute() for an example.
      */
    explicit ComplexSchur(Index size = RowsAtCompileTime == Dynamic ? 1 : RowsAtCompileTime)
        : m_matT(size, size), m_matU(size, size), m_hess(size), m_isInitialized(false), m_matUisUptodate(false), m_maxIters(-1)
    {
    }

    /** \brief Constructor; computes Schur decomposition of given matrix. 
      * 
      * \param[in]  matrix    Square matrix whose Schur decomposition is to be computed.
      * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.
      *
      * This constructor calls compute() to compute the Schur decomposition.
      *
      * \sa matrixT() and matrixU() for examples.
      */
    template <typename InputType>
    explicit ComplexSchur(const EigenBase<InputType>& matrix, bool computeU = true)
        : m_matT(matrix.rows(), matrix.cols()), m_matU(matrix.rows(), matrix.cols()), m_hess(matrix.rows()), m_isInitialized(false), m_matUisUptodate(false),
          m_maxIters(-1)
    {
        compute(matrix.derived(), computeU);
    }

    /** \brief Returns the unitary matrix in the Schur decomposition. 
      *
      * \returns A const reference to the matrix U.
      *
      * It is assumed that either the constructor
      * ComplexSchur(const MatrixType& matrix, bool computeU) or the
      * member function compute(const MatrixType& matrix, bool computeU)
      * has been called before to compute the Schur decomposition of a
      * matrix, and that \p computeU was set to true (the default
      * value).
      *
      * Example: \include ComplexSchur_matrixU.cpp
      * Output: \verbinclude ComplexSchur_matrixU.out
      */
    const ComplexMatrixType& matrixU() const
    {
        eigen_assert(m_isInitialized && "ComplexSchur is not initialized.");
        eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the ComplexSchur decomposition.");
        return m_matU;
    }

    /** \brief Returns the triangular matrix in the Schur decomposition. 
      *
      * \returns A const reference to the matrix T.
      *
      * It is assumed that either the constructor
      * ComplexSchur(const MatrixType& matrix, bool computeU) or the
      * member function compute(const MatrixType& matrix, bool computeU)
      * has been called before to compute the Schur decomposition of a
      * matrix.
      *
      * Note that this function returns a plain square matrix. If you want to reference
      * only the upper triangular part, use:
      * \code schur.matrixT().triangularView<Upper>() \endcode 
      *
      * Example: \include ComplexSchur_matrixT.cpp
      * Output: \verbinclude ComplexSchur_matrixT.out
      */
    const ComplexMatrixType& matrixT() const
    {
        eigen_assert(m_isInitialized && "ComplexSchur is not initialized.");
        return m_matT;
    }

    /** \brief Computes Schur decomposition of given matrix. 
      * 
      * \param[in]  matrix  Square matrix whose Schur decomposition is to be computed.
      * \param[in]  computeU  If true, both T and U are computed; if false, only T is computed.

      * \returns    Reference to \c *this
      *
      * The Schur decomposition is computed by first reducing the
      * matrix to Hessenberg form using the class
      * HessenbergDecomposition. The Hessenberg matrix is then reduced
      * to triangular form by performing QR iterations with a single
      * shift. The cost of computing the Schur decomposition depends
      * on the number of iterations; as a rough guide, it may be taken
      * on the number of iterations; as a rough guide, it may be taken
      * to be \f$25n^3\f$ complex flops, or \f$10n^3\f$ complex flops
      * if \a computeU is false.
      *
      * Example: \include ComplexSchur_compute.cpp
      * Output: \verbinclude ComplexSchur_compute.out
      *
      * \sa compute(const MatrixType&, bool, Index)
      */
    template <typename InputType> ComplexSchur& compute(const EigenBase<InputType>& matrix, bool computeU = true);

    /** \brief Compute Schur decomposition from a given Hessenberg matrix
     *  \param[in] matrixH Matrix in Hessenberg form H
     *  \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
     *  \param computeU Computes the matriX U of the Schur vectors
     * \return Reference to \c *this
     * 
     *  This routine assumes that the matrix is already reduced in Hessenberg form matrixH
     *  using either the class HessenbergDecomposition or another mean. 
     *  It computes the upper quasi-triangular matrix T of the Schur decomposition of H
     *  When computeU is true, this routine computes the matrix U such that 
     *  A = U T U^T =  (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix
     * 
     * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix
     * is not available, the user should give an identity matrix (Q.setIdentity())
     * 
     * \sa compute(const MatrixType&, bool)
     */
    template <typename HessMatrixType, typename OrthMatrixType>
    ComplexSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU = true);

    /** \brief Reports whether previous computation was successful.
      *
      * \returns \c Success if computation was successful, \c NoConvergence otherwise.
      */
    ComputationInfo info() const
    {
        eigen_assert(m_isInitialized && "ComplexSchur is not initialized.");
        return m_info;
    }

    /** \brief Sets the maximum number of iterations allowed. 
      *
      * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size
      * of the matrix.
      */
    ComplexSchur& setMaxIterations(Index maxIters)
    {
        m_maxIters = maxIters;
        return *this;
    }

    /** \brief Returns the maximum number of iterations. */
    Index getMaxIterations() { return m_maxIters; }

    /** \brief Maximum number of iterations per row.
      *
      * If not otherwise specified, the maximum number of iterations is this number times the size of the
      * matrix. It is currently set to 30.
      */
    static const int m_maxIterationsPerRow = 30;

protected:
    ComplexMatrixType m_matT, m_matU;
    HessenbergDecomposition<MatrixType> m_hess;
    ComputationInfo m_info;
    bool m_isInitialized;
    bool m_matUisUptodate;
    Index m_maxIters;

private:
    bool subdiagonalEntryIsNeglegible(Index i);
    ComplexScalar computeShift(Index iu, Index iter);
    void reduceToTriangularForm(bool computeU);
    friend struct internal::complex_schur_reduce_to_hessenberg<MatrixType, NumTraits<Scalar>::IsComplex>;
};

/** If m_matT(i+1,i) is neglegible in floating point arithmetic
  * compared to m_matT(i,i) and m_matT(j,j), then set it to zero and
  * return true, else return false. */
template <typename MatrixType> inline bool ComplexSchur<MatrixType>::subdiagonalEntryIsNeglegible(Index i)
{
    RealScalar d = numext::norm1(m_matT.coeff(i, i)) + numext::norm1(m_matT.coeff(i + 1, i + 1));
    RealScalar sd = numext::norm1(m_matT.coeff(i + 1, i));
    if (internal::isMuchSmallerThan(sd, d, NumTraits<RealScalar>::epsilon()))
    {
        m_matT.coeffRef(i + 1, i) = ComplexScalar(0);
        return true;
    }
    return false;
}

/** Compute the shift in the current QR iteration. */
template <typename MatrixType> typename ComplexSchur<MatrixType>::ComplexScalar ComplexSchur<MatrixType>::computeShift(Index iu, Index iter)
{
    using std::abs;
    if (iter == 10 || iter == 20)
    {
        // exceptional shift, taken from http://www.netlib.org/eispack/comqr.f
        return abs(numext::real(m_matT.coeff(iu, iu - 1))) + abs(numext::real(m_matT.coeff(iu - 1, iu - 2)));
    }

    // compute the shift as one of the eigenvalues of t, the 2x2
    // diagonal block on the bottom of the active submatrix
    Matrix<ComplexScalar, 2, 2> t = m_matT.template block<2, 2>(iu - 1, iu - 1);
    RealScalar normt = t.cwiseAbs().sum();
    t /= normt;  // the normalization by sf is to avoid under/overflow

    ComplexScalar b = t.coeff(0, 1) * t.coeff(1, 0);
    ComplexScalar c = t.coeff(0, 0) - t.coeff(1, 1);
    ComplexScalar disc = sqrt(c * c + RealScalar(4) * b);
    ComplexScalar det = t.coeff(0, 0) * t.coeff(1, 1) - b;
    ComplexScalar trace = t.coeff(0, 0) + t.coeff(1, 1);
    ComplexScalar eival1 = (trace + disc) / RealScalar(2);
    ComplexScalar eival2 = (trace - disc) / RealScalar(2);
    RealScalar eival1_norm = numext::norm1(eival1);
    RealScalar eival2_norm = numext::norm1(eival2);
    // A division by zero can only occur if eival1==eival2==0.
    // In this case, det==0, and all we have to do is checking that eival2_norm!=0
    if (eival1_norm > eival2_norm)
        eival2 = det / eival1;
    else if (eival2_norm != RealScalar(0))
        eival1 = det / eival2;

    // choose the eigenvalue closest to the bottom entry of the diagonal
    if (numext::norm1(eival1 - t.coeff(1, 1)) < numext::norm1(eival2 - t.coeff(1, 1)))
        return normt * eival1;
    else
        return normt * eival2;
}

template <typename MatrixType>
template <typename InputType>
ComplexSchur<MatrixType>& ComplexSchur<MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeU)
{
    m_matUisUptodate = false;
    eigen_assert(matrix.cols() == matrix.rows());

    if (matrix.cols() == 1)
    {
        m_matT = matrix.derived().template cast<ComplexScalar>();
        if (computeU)
            m_matU = ComplexMatrixType::Identity(1, 1);
        m_info = Success;
        m_isInitialized = true;
        m_matUisUptodate = computeU;
        return *this;
    }

    internal::complex_schur_reduce_to_hessenberg<MatrixType, NumTraits<Scalar>::IsComplex>::run(*this, matrix.derived(), computeU);
    computeFromHessenberg(m_matT, m_matU, computeU);
    return *this;
}

template <typename MatrixType>
template <typename HessMatrixType, typename OrthMatrixType>
ComplexSchur<MatrixType>& ComplexSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU)
{
    m_matT = matrixH;
    if (computeU)
        m_matU = matrixQ;
    reduceToTriangularForm(computeU);
    return *this;
}
namespace internal {

    /* Reduce given matrix to Hessenberg form */
    template <typename MatrixType, bool IsComplex> struct complex_schur_reduce_to_hessenberg
    {
        // this is the implementation for the case IsComplex = true
        static void run(ComplexSchur<MatrixType>& _this, const MatrixType& matrix, bool computeU)
        {
            _this.m_hess.compute(matrix);
            _this.m_matT = _this.m_hess.matrixH();
            if (computeU)
                _this.m_matU = _this.m_hess.matrixQ();
        }
    };

    template <typename MatrixType> struct complex_schur_reduce_to_hessenberg<MatrixType, false>
    {
        static void run(ComplexSchur<MatrixType>& _this, const MatrixType& matrix, bool computeU)
        {
            typedef typename ComplexSchur<MatrixType>::ComplexScalar ComplexScalar;

            // Note: m_hess is over RealScalar; m_matT and m_matU is over ComplexScalar
            _this.m_hess.compute(matrix);
            _this.m_matT = _this.m_hess.matrixH().template cast<ComplexScalar>();
            if (computeU)
            {
                // This may cause an allocation which seems to be avoidable
                MatrixType Q = _this.m_hess.matrixQ();
                _this.m_matU = Q.template cast<ComplexScalar>();
            }
        }
    };

}  // end namespace internal

// Reduce the Hessenberg matrix m_matT to triangular form by QR iteration.
template <typename MatrixType> void ComplexSchur<MatrixType>::reduceToTriangularForm(bool computeU)
{
    Index maxIters = m_maxIters;
    if (maxIters == -1)
        maxIters = m_maxIterationsPerRow * m_matT.rows();

    // The matrix m_matT is divided in three parts.
    // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero.
    // Rows il,...,iu is the part we are working on (the active submatrix).
    // Rows iu+1,...,end are already brought in triangular form.
    Index iu = m_matT.cols() - 1;
    Index il;
    Index iter = 0;       // number of iterations we are working on the (iu,iu) element
    Index totalIter = 0;  // number of iterations for whole matrix

    while (true)
    {
        // find iu, the bottom row of the active submatrix
        while (iu > 0)
        {
            if (!subdiagonalEntryIsNeglegible(iu - 1))
                break;
            iter = 0;
            --iu;
        }

        // if iu is zero then we are done; the whole matrix is triangularized
        if (iu == 0)
            break;

        // if we spent too many iterations, we give up
        iter++;
        totalIter++;
        if (totalIter > maxIters)
            break;

        // find il, the top row of the active submatrix
        il = iu - 1;
        while (il > 0 && !subdiagonalEntryIsNeglegible(il - 1)) { --il; }

        /* perform the QR step using Givens rotations. The first rotation
       creates a bulge; the (il+2,il) element becomes nonzero. This
       bulge is chased down to the bottom of the active submatrix. */

        ComplexScalar shift = computeShift(iu, iter);
        JacobiRotation<ComplexScalar> rot;
        rot.makeGivens(m_matT.coeff(il, il) - shift, m_matT.coeff(il + 1, il));
        m_matT.rightCols(m_matT.cols() - il).applyOnTheLeft(il, il + 1, rot.adjoint());
        m_matT.topRows((std::min)(il + 2, iu) + 1).applyOnTheRight(il, il + 1, rot);
        if (computeU)
            m_matU.applyOnTheRight(il, il + 1, rot);

        for (Index i = il + 1; i < iu; i++)
        {
            rot.makeGivens(m_matT.coeffRef(i, i - 1), m_matT.coeffRef(i + 1, i - 1), &m_matT.coeffRef(i, i - 1));
            m_matT.coeffRef(i + 1, i - 1) = ComplexScalar(0);
            m_matT.rightCols(m_matT.cols() - i).applyOnTheLeft(i, i + 1, rot.adjoint());
            m_matT.topRows((std::min)(i + 2, iu) + 1).applyOnTheRight(i, i + 1, rot);
            if (computeU)
                m_matU.applyOnTheRight(i, i + 1, rot);
        }
    }

    if (totalIter <= maxIters)
        m_info = Success;
    else
        m_info = NoConvergence;

    m_isInitialized = true;
    m_matUisUptodate = computeU;
}

}  // end namespace Eigen

#endif  // EIGEN_COMPLEX_SCHUR_H
